# Properties

 Modulus $156$ Structure $$C_{12}\times C_{2}\times C_{2}$$ Order $48$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(156)

pari: g = idealstar(,156,2)

## Character group

 sage: G.order()  pari: g.no Order = 48 sage: H.invariants()  pari: g.cyc Structure = $$C_{12}\times C_{2}\times C_{2}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{156}(79,\cdot)$, $\chi_{156}(53,\cdot)$, $\chi_{156}(145,\cdot)$

## First 32 of 48 characters

Each row describes a character. When available, the columns show the orbit label, order of the character, whether the character is primitive, and several values of the character.

Character Orbit Order Primitive $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$ $$35$$
$$\chi_{156}(1,\cdot)$$ 156.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{156}(5,\cdot)$$ 156.m 4 no $$1$$ $$1$$ $$i$$ $$i$$ $$-i$$ $$1$$ $$-i$$ $$1$$ $$-1$$ $$-1$$ $$-i$$ $$-1$$
$$\chi_{156}(7,\cdot)$$ 156.w 12 no $$1$$ $$1$$ $$i$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$-i$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{156}(11,\cdot)$$ 156.v 12 yes $$-1$$ $$1$$ $$-i$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$-i$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{156}(17,\cdot)$$ 156.s 6 no $$-1$$ $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{156}(19,\cdot)$$ 156.w 12 no $$1$$ $$1$$ $$-i$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$i$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{156}(23,\cdot)$$ 156.r 6 yes $$1$$ $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{156}(25,\cdot)$$ 156.b 2 no $$1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$
$$\chi_{156}(29,\cdot)$$ 156.o 6 no $$-1$$ $$1$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{156}(31,\cdot)$$ 156.k 4 no $$1$$ $$1$$ $$-i$$ $$-i$$ $$-i$$ $$-1$$ $$i$$ $$1$$ $$-1$$ $$1$$ $$i$$ $$-1$$
$$\chi_{156}(35,\cdot)$$ 156.p 6 yes $$1$$ $$1$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{156}(37,\cdot)$$ 156.x 12 no $$-1$$ $$1$$ $$i$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$i$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{156}(41,\cdot)$$ 156.u 12 no $$1$$ $$1$$ $$i$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$-i$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{156}(43,\cdot)$$ 156.n 6 no $$-1$$ $$1$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{156}(47,\cdot)$$ 156.l 4 yes $$-1$$ $$1$$ $$-i$$ $$i$$ $$-i$$ $$1$$ $$-i$$ $$-1$$ $$-1$$ $$-1$$ $$-i$$ $$1$$
$$\chi_{156}(49,\cdot)$$ 156.q 6 no $$1$$ $$1$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{156}(53,\cdot)$$ 156.d 2 no $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$
$$\chi_{156}(55,\cdot)$$ 156.t 6 no $$-1$$ $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{156}(59,\cdot)$$ 156.v 12 yes $$-1$$ $$1$$ $$-i$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$-i$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{156}(61,\cdot)$$ 156.i 3 no $$1$$ $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{156}(67,\cdot)$$ 156.w 12 no $$1$$ $$1$$ $$-i$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$i$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{156}(71,\cdot)$$ 156.v 12 yes $$-1$$ $$1$$ $$i$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$i$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{156}(73,\cdot)$$ 156.j 4 no $$-1$$ $$1$$ $$i$$ $$-i$$ $$-i$$ $$-1$$ $$i$$ $$-1$$ $$-1$$ $$1$$ $$i$$ $$1$$
$$\chi_{156}(77,\cdot)$$ 156.g 2 no $$-1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$
$$\chi_{156}(79,\cdot)$$ 156.f 2 no $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$
$$\chi_{156}(83,\cdot)$$ 156.l 4 yes $$-1$$ $$1$$ $$i$$ $$-i$$ $$i$$ $$1$$ $$i$$ $$-1$$ $$-1$$ $$-1$$ $$i$$ $$1$$
$$\chi_{156}(85,\cdot)$$ 156.x 12 no $$-1$$ $$1$$ $$i$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$i$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{156}(89,\cdot)$$ 156.u 12 no $$1$$ $$1$$ $$-i$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$i$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{156}(95,\cdot)$$ 156.r 6 yes $$1$$ $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{156}(97,\cdot)$$ 156.x 12 no $$-1$$ $$1$$ $$-i$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$-i$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{156}(101,\cdot)$$ 156.s 6 no $$-1$$ $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{156}(103,\cdot)$$ 156.e 2 no $$-1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$-1$$